Analysis of the interaction between an incompressible fluid and a quasi-incompressible non-linear elastic structure

TL;DR

This paper investigates a complex fluid-structure interaction involving an incompressible fluid and a quasi-incompressible nonlinear elastic structure, establishing local existence and uniqueness of solutions using linearization and fixed point methods.

Contribution

It introduces a novel approach to prove local well-posedness for a coupled Navier-Stokes and quasi-incompressible elastic model in three dimensions.

Findings

Proved local in time existence and uniqueness of solutions.

Established a method to handle non-linear elastodynamics explicitly.

Demonstrated the existence of fluid pressure via the inf-sup condition.

Abstract

We are interested in studying an unsteady fluid-structure interaction problem in a three-dimensional space. We consider a homogeneous Newtonian fluid which is modeled by the Navier-Stokes equations. Whereas the motion of the structure is described by the quasi-incompressible non-linear Saint Venant-Kirchhoff model. We establish the local in time existence and uniqueness of solution for this model. For this sake, first we rewrite the non-linearity of the elastodynamic equation in an explicit way. Then, a linearized problem is introduced in the Lagrangian reference configuration and we prove that it admits a unique solution. Based on the a priori estimates on the solution of this problem together with the fixed point theorem we prove that the non-linear problem admits a unique local in time solution. At last, by the inf-sup condition we reach to the existence of the fluid pressure.